Directivity is one of the core performance parameters for an antenna, and its accurate computation and measurement have been receiving attention in the past decades. While quite a number of computational methods for antenna directivity are developed, a comprehensive comparison among these methods, including their advantages and disadvantages, has not yet been reported. In the literature, most of these methods can be roughly classified as two categories, i.e. numerical methods and analytical methods. The numerical methods include those simplified numerical methods for accessing the radiation pattern. For example, some methods assume that the radiation pattern is only elevation-angle dependent, and an approximate truncation of the integral interval is often employed. The analytical methods are developed, where the electromagnetic radiation intensity and the total radiation power for a specific antenna are derived analytically. Nevertheless, so far there have been less efforts dedicated to developing general computational methods based on the antenna's planar near-field theory and associated computational schemes, although indeed quite a few of advanced mathematical approaches have been used to obtain the accurate directivity, including the series expansion based on the spherical wave scheme and the Fourier expansion scheme. In spite of these efforts, one can see that a general accurate numerical scheme for the antenna directivity is still urgently required. As an important supplement to these efforts, this paper adopts or proposes four numerical integration methods regarding planar near-field measurement of the antennas under test. Based on the planar near-field theory, microwave near-field data, and fast Fourier transform (FFT), this paper discusses the numerical integration-based algorithm schemes based on the simulated (or measured) near-field data and relevant numerical uncertainty estimation. Specifically, an antenna with accurate directivity is chosen for illustrating the accuracy assessment. First, a set of near-field data under the well-established probing condition is simulated and then used as an input for subsequent numerical calculation of the antenna directivity value. Four different numerical integration methods are employed and their possible numerical errors are discussed. Consequently, the antenna directivity values are obtained respectively from the four different numerical methods. It is revealed that the numerically evaluated directivity values from the near-field data are roughly at most 0.6 dB larger than the accurate value. This work represents a substantial step toward a reliable estimation of the antenna directivity from the near-field data of an antenna.